What is the general solution for 4sin^2(x)+3sin(x)-1=0 ?

The general solution for 4sin^2(x)+3sin(x)-1=0 is x=0.25268…+2pin,x=pi-0.25268…+2pin,x=(3pi)/2+2pin

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4sin^2(x)+3sin(x)-1=0

4 sin^{2} (x) + 3 sin (x) - 1 = 0

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

Solution steps

4 sin^{2} (x) + 3 sin (x) - 1 = 0

Solve by substitution

sin (x) = \frac{1}{4}, sin (x) = - 1

sin (x) = \frac{1}{4} : x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn, x = \frac{3 π}{2} + 2 πn

Show solutions in decimal form

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = \frac{π}{2}

2 cos (θ) + 2 = 0

2 sin (x) + cos (x) = 0

sin (x + \frac{π}{2}) + sin (x - \frac{3 π}{2}) = 1

- \frac{1}{2} = cos (x)

What is the general solution for 4sin^2(x)+3sin(x)-1=0 ?
The general solution for 4sin^2(x)+3sin(x)-1=0 is x=0.25268…+2pin,x=pi-0.25268…+2pin,x=(3pi)/2+2pin